I am a little stuck in seeing how the partial sums of an alternating Harmonic series satisfy the cauchy condition.

i.e. The partial sums are defined as

This yields for

Its this last inequality I am having trouble verifying. Any ideas?

December 11th 2009, 11:31 PM

tonio

Quote:

Originally Posted by aukie

Hello

I am a little stuck in seeing how the partial sums of an alternating Harmonic series satisfy the cauchy condition.

i.e. The partial sums are defined as

This yields for

Its this last inequality I am having trouble verifying. Any ideas?

You can try induction on :

First, it is not hard to see that .

Now we have two cases: either , or .
In the first case it is trivial, whereas in the second case you can use inductive hypothesis not with m-1 but with m-2, since

if , then

Tonio

December 12th 2009, 01:59 AM

aukie

Hello Tonio

In working out why the absolute value can be ommitted and what you meant by trivial. I seen an argument that doesn't require induction. If you're interested: Assuming m = n + k,

Group the terms as follows (for even k)

(and for odd k)

Then we have a series of positive terms subtracted from so

Thanks for your input

December 12th 2009, 07:31 AM

tonio

Quote:

Originally Posted by aukie

Hello Tonio

In working out why the absolute value can be ommitted and what you meant by trivial. I seen an argument that doesn't require induction. If you're interested: Assuming m = n + k,

Group the terms as follows (for even k)

(and for odd k)

Then we have a series of positive terms subtracted from so

Thanks for your input

I think it works just fine! Yet even in this case we have what's sometimes called "concealed or hidden" induction: it is a weak form and it just could be stressed that what you did actually works no matter what in you used...